Musical scales

All such scales can be uniquely described by the set of intervals between consecutive (in the scale) notes. For example, the major scale is characterized by intervals of $2, 2, 1, 2, 2, 2, 1$ semitones (ST)

To satisfy the conditions, the sum of two consecutive intervals has to be either 3 (minor third) or 4 (major third) ST and the sum of all intervals must be 12 ST to repeat after one octave. This makes the following combinations of intervals possible.

$\begin{array}{ccccc} + & 1 & 2 & 3 & 1^{\text{st}} \\ 1 & 2(-) & 3(+) & 4(+) \\ 2 & 3(+) & 4(+) & 5(-) \\ 3 & 4(+) & 5(-) & 6(-) \\ 2^{\text{nd}} &&&& \end{array}$

Now, we split the problem up into cases:

Case 1: Intervals of 1, 2 and 3 ST

The 3 ST interval has to be after and before a 1 ST interval which already creates 5 ST. If this was followed by another 3 ST interval, there wouldn't be any space left for 2 ST intervals.

Therefore, we get the beginning 2 1 3 1 2 which can either end in 1 2 or 2.

Of course, we could start at any point at this scale, so for both options there are 7 different starting points for a total of $2 \cdot 7 = 14$ scales.

Case 2: no 1 ST interval

Since we can only have a 3 ST interval in combination with 1 ST intervals, we can actually only use 2 ST intervals, so the only scale is 2 2 2 2 2 2.

Case 3: no 2 ST interval

We start like case 1, but this time, we keep adding 3s and 1s until the octave is full. Thid leads to 1 3 1 3 1 3 and 3 1 3 1 3 1.

Case 4: no 3 ST interval

Since the sum of all intervals must be 12 ST there can only be an even number of 1 ST intervals, either 4 or 2 (6 ist to much and we want at least one)

Case 4.1: 4 intervals of 1 ST

We have to have 4 intervals of 2 ST, so the only way to meet thr criterea from the table is by alternating them, as in 1 2 1 2 1 2 1 2 or 2 1 2 1 2 1 2 1.

Case 4.2: 2 intervals of 1 ST

There will be 5 intervals of 2 ST, so 7 intervals in total. Of those, we can choose one freely, but the second one can't be the same and the one before or after it. This makes $7 \cdot 4 = 28$ combinations, but we've counted all twice so there are $28 : 2 = 14$ scales.

Overall, we get $14 + 1 + 2 + 2 + 14 = \boxed{33}$ such scales.